Question

Why $p^q\Rightarrow r$ is true when p is true and q false.

Answer 1

If we have predicts $p$ and $q$ then $p\Rightarrow q$ is true when either $p$ is false or $q$ is true (or both) that is

$\left(p\Rightarrow q\right)\iff \left(-pvq\right)$

The implication will not hold only when $p$ is true and $q$ is false. that is $p^{-q}$ is true.

So in question, $p$ is true and $q$ is false.

So, $p^q$ is false. thus $p^q\Rightarrow r$ is indeed true.

$\left(p^q\Rightarrow r\right)\iff \left(-\left(p^q\right)vr\right)$